Mesh refinement

Abstract

A computer-implemented method comprising: generating a main graph based on a coarse mesh, the coarse mesh comprising a plurality of mesh elements arranged to fill an area or volume defined by a geometry; generating a vertex graph based on the geometry, wherein the vertex graph comprises vertex graph nodes corresponding to vertices of the area or volume defined by the geometry and vertex graph edges corresponding to links between the vertices in the geometry; generating, using a first graph neural network (GNN), an embedding of the vertex graph; generating, using a second GNN and based on the main graph and the embedding of the vertex graph, a prediction indicative of a refinement of the coarse mesh for generating a refined mesh.

1 . A computer-implemented method comprising:
 generating a main graph based on a coarse mesh, the coarse mesh comprising a plurality of mesh elements arranged to fill an area or volume defined by a geometry;   generating a vertex graph based on the geometry, wherein the vertex graph comprises vertex graph nodes corresponding to vertices of the area or volume defined by the geometry and vertex graph edges corresponding to links between the vertices in the geometry;   generating, using a first graph neural network (GNN), an embedding of the vertex graph;   generating, using a second GNN and based on the main graph and the embedding of the vertex graph, a prediction indicative of a refinement of the coarse mesh for generating a refined mesh.   
     
     
         2 . The computer-implemented method as claimed in  claim 1 , wherein the prediction comprises a predicted error and wherein the computer-implemented method further comprises comparing the predicted error to a ground truth error and adjusting at least one weight of the first and/or second GNN based on the comparison, wherein the ground truth error comprises an error between a simulation result obtained using the coarse mesh and a simulation result obtained using a fine mesh. 
     
     
         3 . The computer-implemented method as claimed in  claim 2 , wherein the first and second GNNs after having at least one weight adjusted are trained first and second GNNs, and wherein the computer-implemented method further comprises:
 generating a target main graph based on a target coarse mesh, the target coarse mesh comprising a plurality of mesh elements arranged to fill an area or volume defined by a geometry;   generating a target vertex graph based on the geometry, wherein the target vertex graph comprises vertex graph nodes corresponding to vertices of the area or volume defined by the geometry and vertex graph edges corresponding to links between the vertices in the geometry;   generating, using the trained first GNN, an embedding of the target vertex graph;   generating, using the trained second GNN and based on the target main graph and the embedding of the target vertex graph, a prediction indicative of a refinement of the target coarse mesh for generating a refined mesh; and   refining the target coarse mesh based on the prediction to obtain the refined mesh.   
     
     
         4 . The computer-implemented method as claimed in  claim 1 , further comprising refining the coarse mesh based on the prediction to obtain a refined mesh. 
     
     
         5 . The computer-implemented method according to  claim 3 , comprising computing a simulation using the refined mesh. 
     
     
         6 . The computer-implemented method as claimed in  claim 3 , further comprising using the refined mesh to model a physical component. 
     
     
         7 . The computer-implemented method according to  claim 3 , comprising using the refined mesh to model a physical component of any of a building, an automobile, a biomechanical or physiological system, and an aircraft. 
     
     
         8 . The computer-implemented method as claimed  claim 1 , wherein the coarse mesh comprises coarse mesh nodes and coarse mesh edges defining the elements of the coarse mesh and wherein generating the vertex graph comprises generating a per-node vertex graph per coarse mesh node, wherein each per-node vertex graph comprises a plurality of per-node vertex graph nodes corresponding respectively to the vertices. 
     
     
         9 . The computer-implemented method as claimed in  claim 8 , wherein each per-node vertex graph node comprises geometric information relating to the coarse mesh node corresponding to the per-node vertex graph and the vertex corresponding to the per-node vertex graph node, and wherein the geometric information comprises at least one of:
 mean value coordinates of the vertex concerned for the coarse mesh node concerned;   a number of hops between the vertex concerned and the coarse mesh node concerned;   the vertex closest to the coarse mesh node concerned among the vertices;   boundary condition information for the vertex concerned;   parameters related to a simulation for which the refined mesh is to be used.   
     
     
         10 . The computer-implemented method as claimed in  claim 1 , wherein the first GNN comprises at least one multi-layer perceptron layer and wherein the second GNN comprises a residual-layers-based architecture. 
     
     
         11 . The computer-implemented method as claimed in  claim 10 , wherein the second GNN comprises at least one multi-layer perceptron layer. 
     
     
         12 . The computer-implemented method as claimed in  claim 2 , wherein comparing the predicted error to the ground truth error comprises computing a loss between on the predicted error and the ground truth error, and wherein adjusting the at least one weight of the first and/or second GNN based on the comparison comprises adjusting the at least one weight to reduce the loss. 
     
     
         13 . The computer-implemented method as claimed in  claim 12 , wherein computing the loss comprises computing the L 1  loss function between the predicted error and the ground truth error. 
     
     
         14 . The computer-implemented method as claimed in  claim 12 , wherein the prediction comprises a per-node predicted error per coarse mesh node of the coarse mesh. 
     
     
         15 . The computer-implemented method as claimed in  claim 14 , wherein the ground truth error comprises a per-node ground truth error per coarse mesh node of the coarse mesh. 
     
     
         16 . The computer-implemented method as claimed in  claim 15 , wherein computing the loss comprises computing an average of per-node losses between corresponding per-node predicted errors and per-node ground truth errors. 
     
     
         17 . The computer-implemented method as claimed in  claim 2 , comprising obtaining the ground truth error by performing a simulation using the coarse mesh and performing a simulation using the fine mesh, and comparing results of the simulations to determine the error between the results of the simulations. 
     
     
         18 . The computer-implemented method as claimed in  claim 2 , comprising iterating the comparing the predicted error to the ground truth error and the adjusting at least one weight of the first and/or second GNN until the comparison indicates a loss threshold is met or until the comparison indicates loss convergence or until a threshold number of iterations have been performed. 
     
     
         19 . A computer program which, when run on a computer, causes the computer to carry out a method comprising:
 generating a main graph based on a coarse mesh, the coarse mesh comprising a plurality of mesh elements arranged to fill an area or volume defined by a geometry;   generating a vertex graph based on the geometry, wherein the vertex graph comprises vertex graph nodes corresponding to vertices of the area or volume defined by the geometry and vertex graph edges corresponding to links between the vertices in the geometry;   generating, using a first graph neural network (GNN), an embedding of the vertex graph;   generating, using a second GNN and based on the main graph and the embedding of the vertex graph, a prediction indicative of a refinement of the coarse mesh for generating a refined mesh.   
     
     
         20 . An information processing apparatus comprising a memory and a processor connected to the memory, wherein the processor is configured to:
 generate a main graph based on a coarse mesh, the coarse mesh comprising a plurality of mesh elements arranged to fill an area or volume defined by a geometry;   generate a vertex graph based on the geometry, wherein the vertex graph comprises vertex graph nodes corresponding to vertices of the area or volume defined by the geometry and vertex graph edges corresponding to links between the vertices in the geometry;   generate, using a first graph neural network (GNN), an embedding of the vertex graph;   generate, using a second GNN and based on the main graph and the embedding of the vertex graph, a prediction indicative of a refinement of the coarse mesh for generating a refined mesh.